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Electromagnetic Fields

Parallel Plate Waveguide

w
a

x
0

z
y

a

Assume uniform waves along the y-direction Assume no fringing effects

⇒ w >> a

∂ ⇒ ( ∂y

)=0

Propagation along the z-direction
© Amanogawa, 2006 – Digital Maestro Series 130

Electromagnetic Fields

Maxwell’s equations

∇ × E = − jω µ H ⇓ ˆ ˆ ˆ  ix i y iz    ∂ ∂   ∂ det ∂ x ∂ y ∂ z  Ex E y Ez    ∂ ∂ E z − E y = − jωµH x ∂y ∂z ⇒ (1)

∂ ∂ E x − E z = − jωµH y (2) ∂z ∂x ∂ ∂ E y − E x = − jω µH z ∂x ∂y (3)

© Amanogawa, 2006 – Digital Maestro Series

131

Electromagnetic Fields

∇ × H = jω ε E ⇓ ˆ ˆ ˆ  ix iy iz    ∂ ∂   ∂ det ∂ x ∂ y ∂ z   H x H y H z    ∂ ∂ H z − H y = jωεE x ∂y ∂z ⇒ (4)

∂ ∂ H x − H z = jωεE y (5) ∂z ∂x ∂ ∂ H y − H x = jωεE z ∂y ∂x(6)

© Amanogawa, 2006 – Digital Maestro Series

132

Electromagnetic Fields

From (1) & (2) & (5)

∂ (1) ∂z

∂ ⇒ E y = jωµ H x ∂z ∂ z2 ∂ E = − jω µ H z 2 y ∂x ∂x ∂2

∂2

∂ (3) ⇒ ∂x ∂2 ∂2

∂  ∂  Ey + E y = jωµ  H x − H z  = −ω2µ ε E y ∂x  ∂z  ∂ z2 ∂ x2 From (5) ⇓ jωε E y

Wave equation for Transverse Electric (TE) modes
© Amanogawa, 2006 – Digital Maestro Series 133 Electromagnetic Fields

From (4) & (6) & (2)

∂ (4) ∂z

∂ H y = − jωε E x ∂z ∂ z2 ∂ H = jωε E z 2 y ∂x ∂x ∂2

∂2

∂ (6) ⇒ ∂x ∂2 ∂2

∂  ∂  Hy + H y = − jωε  E x − E z  = −ω2µ ε H y ∂x   ∂z ∂ z2 ∂ x2 From (2) ⇓ − jωµ H y

Wave equation for Transverse Magnetic (TM) modes
© Amanogawa, 2006 – Digital Maestro Series 134

Electromagnetic Fields

Transverse Electric (TE)modes

E H

β

θ θ

H
×

E
θ θ

β
H

E

Boundary Conditions

Ey = 0

x = 0  x = a

This solution satisfies the boundary conditions:

E y = Eo sin ( β x x ) e

− jβ z z

Eo − jβ x x − jβ z = j e − e jβ x x e z 2
x z
135

(

)

© Amanogawa, 2006 – Digital Maestro Series

Electromagnetic Fields

We have

β2 =

4π2 λ2

= β 2 + β 2 = ω 2µ ε x z

andfrom boundary conditions at the conductor plates

x = 0) E y = 0 x = a) sin ( β x a ) = 0 ⇒ β x a = m π m = 1, 2, 3… mπ β x = β cos θ = a   mλ   mπ  β z = β sin θ = ω µ ε −    = ω µ ε 1 −    2a    a    
2 2
© Amanogawa, 2006 – Digital Maestro Series

2 1 / 2

136

Electromagnetic Fields

For each possible index m we have a mode of propagation. Modes are labeled TE10, TE20 , TE30 , …. The first index gives the periodicity (number of half sinusoidal oscillations) between the plates, along the x-direction. The second index is zero to indicate uniform solution along the y-direction. Note that the solution m = 0 (or mode TE00) is not acceptable, because it would require a field configuration with uniform electric field tangent to the metal plates. This is anunphysical boundary condition, which is possible only for the case of trivial solution of zero field everywhere.

E H

β

TE00 ⇒ m = 0 ⇒ βx = 0 & β = βz Unphysical !!!
137

© Amanogawa, 2006 – Digital Maestro Series

Electromagnetic Fields

A mode can propagate only if the frequency is sufficiently high, so that βz > 0. We have the cut-off condition when

mπ 2 π 2a β = βx = = ⇒ λc = aλc m
2   mλ  2  2 mπ   ⇒ β z = ω2 µ ε −   = ω µ ε 1 −    =0   2a    a    vp m fc = Cut - off frequency for mode m = λ c 2 a µε
Exactly at cut-off the wave would bounce between the plates, without propagation along the wave guide axis.
© Amanogawa, 2006 – Digital Maestro Series 138

1

Electromagnetic Fields

When the frequency is below the cut-off value

2a f < fc ⇒λ > λ c = m  mπ  2 βz = ± ω µ ε −    a 
2

  2   mλ  2  = ±ω µ ε  1 −      2a      >1 
1

1

  mλ  2  2 = ± j ω µε   − 1   2a    ⇒ β z = − jα ⇒ e− j( − jα ) z = e−αz
The mode attenuates entering the guide as an evanescent wave.
© Amanogawa, 2006 – Digital Maestro Series 139

Electromagnetic Fields

Transverse Magnetic (TM) modes

E

β...